3 Strategy of interpretation

For those who adhere to the principle of evolution, the pivotal role played by measurements in standard formulations of quantum mechanics is an embarrassment known as the “measurement problem.” As an anonymous referee for a philosophy of science journal once put it to me, “to solve this problem means to design an interpretation in which measurement processes are not different in principle from ordinary physical interactions.” How can reducing measurements to “ordinary physical interactions” solve this problem, considering that quantum mechanics describes “ordinary physical interactions” in terms of correlations between the probabilities of the possible outcomes of measurements performed on the interacting systems? This kind of “solution” merely sweeps the problem under the rug.

In actual fact, to solve the measurement problem means to design an interpretation in which the central role played by measurements in standard axiomatizations of quantum mechanics is understood. But before there is any hope of understanding it, the obvious must be acknowledged — namely, that the formalism of quantum mechanics is a probability calculus, and that the events to which this assigns probabilities are measurement outcomes.

An algorithm for assigning probabilities to possible measurement outcomes on the basis of actual outcomes has two perfectly normal dependencies. It depends continuously on the time of measurement: if this changes by a small amount, the assigned probabilities change by small amounts. And it depends discontinuously on the outcomes that constitute the assignment basis: if this changes by the inclusion of an outcome not previously taken into account, so do the assigned probabilities.

But think of a quantum state’s dependence on time as the continuous time-dependence of an evolving state (rather than as a dependence on the time of a measurement), and you have two modes of evolution for the price of one:

between measurements, a quantum state evolves according to a unitary transformation and thus continuously and predictably;

at the time of a measurement, a quantum state generally “collapses” (or appears to “collapse”): it changes (or appears to change) discontinuously and unpredictably into a state that depends on the measurement’s outcome.

Hence the mother of all quantum-theoretical pseudo-questions: why does a quantum state have (or appear to have) two modes of evolution? And hence the embarrassment. Getting rid of the pseudo-question is easy: we only have to recognize that the true number of modes of evolution is neither two nor one but zero. Getting rid of the embarrassment requires more work, for we still have two Rules. Why two? What distinguishes this question from the aforementioned pseudo-question is that it has a straightforward answer:

Whenever quantum mechanics instructs us to add the amplitudes of alternatives rather than their probabilities (that is, whenever we are required to use Rule B rather than Rule A), the distinctions we make between the alternatives correspond to nothing in the actual world. They don’t exist in the actual world. They exist solely in our minds.

This answer lies at the heart of the interpretational strategy we shall adopted. It does raise further questions, but it also makes it possible to answer them. What is more, it does not appeal to untestable metaphysical assumptions about what happens between measurements but proceeds directly from the testable computational rules of quantum mechanics.